Chapter VI: Measurement of Belief

§§ 1, 2. Preliminary remarks.
3, 4. Are we accurately conscious of gradations of belief?
5. Probability only concerned with part of this enquiry.
6. Difficulty of measuring our belief;
7. Owing to intrusion of emotions,
8. And complexity of the evidence.
9. And when measured, is it always correct?
10, 11. Distinction between logical and psychological views.
12-16. Analogy of Formal Logic fails to show that we can thus detach
and measure our belief.
17. Apparent evidence of popular language to the contrary.
18. How is full belief justified in inductive enquiry?
19-23. Attempt to show how partial belief may be similarly justified.
24-28. Extension of this explanation to cases which cannot be repeated
in experience.
29. Can other emotions besides belief be thus measured?
30. Errors thus arising in connection with the Petersburg Problem.
31, 32. The emotion of surprise is a partial exception.
33, 34. Objective and subjective phraseology.
35. The definition of probability,
36. Introduces the notion of a 'limit',
37. And implies, vaguely, some degree of belief.

CHAPTER VII.

THE RULES OF INFERENCE IN PROBABILITY.

§ 1. Nature of these inferences.
2. Inferences by addition and subtraction.
3. Inferences by multiplication and division.
4-6. Rule for independent events.
7. Other rules sometimes introduced.
8. All the above rules may be interpreted subjectively, i.e. in terms
of belief.
9-11. Rules of so-called Inverse Probability.
12, 13. Nature of the assumption involved in them:
14-16. Arbitrary character of this assumption.
17, 18. _Physical illustrations._

CHAPTER VIII.

THE RULE OF SUCCESSION.

§ 1. Reasons for desiring some such rule:
2. Though it could scarcely belong to Probability.
3. Distinction between Probability and Induction.
4, 5. Impossibility of reducing the various rules of the latter under
one head.
6. Statement of the Rule of Succession;
7. _Proof offered for it._
8. _Is it a strict rule of inference?_
9. _Or is it a psychological principle?_

CHAPTER IX.

INDUCTION.

§§ 1-5. Statement of the Inductive problem, and origin of the Inductive
inference.
6. Relation of Probability to Induction.
7-9. The two are sometimes merged into one.
10. Extent to which causation is needed in Probability.
11-13. Difficulty of referring an individual to a class:
14. This difficulty but slight in Logic,
15, 16. But leads to perplexity in Probability:
17-21. Mild form of this perplexity;
22, 23. Serious form.
24-27. Illustration from Life Insurance.
28, 29. Meaning of 'the value of a life'.
30, 31. Successive specialization of the classes to which objects are
referred.
32. Summary of results.

CHAPTER X.

CHANGE, CAUSATION AND DESIGN.

§ 1. Old Theological objection to Chance.
2-4. Scientific version of the same.
5. Statistics in reference to Free-will.
6-8. Inconclusiveness of the common arguments here.
9, 10. _Chance as opposed to Physical Causation._
11. _Chance as opposed to Design in the case of numerical constants._
12-14. _Theoretic solution between Chance and Design._
15. _Illustration from the dimensions of the Pyramid._
16, 17. _Discussion of certain difficulties here._
18, 19. _Illustration from Psychical Phenomena._
20. Arbuthnott's Problem of the proportion of the sexes.
21-23. Random or designed distribution of the stars.

(_Note on the proportion of the sexes_.)

CHAPTER XI.

MATERIAL AND FORMAL LOGIC.

§§ 1, 2. _Broad distinction between these views;_
2, 3. _Difficulty of adhering consistently to the objective view;_
4. _Especially in the case of Hypotheses._
5. The doubtful stage of our facts is only occasional in Inductive
Logic.
6-9. But normal and permanent in Probability.
10, 11. Consequent difficulty of avoiding Conceptualist phraseology.

CHAPTER XII.

CONSEQUENCES OF THE DISTINCTIONS OF THE PREVIOUS CHAPTER.

§§ 1, 2. Probability has no relation to time.
3, 4. Butler and Mill on Probability before and after the event.
5. Other attempts at explaining the difficulty.
6-8. What is really meant by the distinction.
9. Origin of the common mistake.
10-12. Examples in illustration of this view,
13. Is Probability relative?
14. What is really meant by this expression.
15. Objections to terming Probability relative.
16, 17. In suitable examples the difficulty scarcely presents itself.

CHAPTER XIII.

ON MODALITY.

§ 1. Various senses of Modality;
2. Having mostly some relation to Probability.
3. Modality must be recognized.
4. Sometimes relegated to the predicate,
5, 6. Sometimes incorrectly rejected altogether.
7, 8. Common practical recognition of it.
9-11. Modal propositions in Logic and in Probability.
12. Aristotelian view of the Modals;
13, 14. Founded on extinct philosophical views;
15. But long and widely maintained.
16. Kant's general view.
17-19. The number of modal divisions admitted by various logicians.
20. Influence of the theory of Probability.
21, 22. Modal syllogisms.
23. Popular modal phraseology.
24-26. Probable and Dialectic syllogisms.
27, 28. Modal difficulties occur in Jurisprudence.
29, 30. Proposed standards of legal certainty.
31. Rejected formally in English Law, but possibly recognized practically.
32. How, if so, it might be determined.

CHAPTER XIV. FALLACIES.

§§ 1-3. (I.) Errors in judging of events after they have happened.
4-7. Very various judgments may be thus involved.
8, 9. (II.) _Confusion between random and picked selections._
10, 11. (III.) Undue limitation of the notion of Probability.
12-16. (IV.) Double or Quits: the Martingale.
17, 18. Physical illustration.
19, 20. (V.) Inadequate realization of large numbers.
21-24. Production of works of art by chance.
25. Illustration from doctrine of heredity.
26-30. (VI.) Confusion between Probability and Induction.
31-33. (VII.) Undue neglect of small chances.
34, 35. (VIII.) _Judging by the event in Probability and in Induction._